論擴散-競爭型洛特卡-佛爾特拉方程組的行進波解之N型屏障最大值原理

碩⼠論⽂

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論擴散@競爭型洛特卡@佛爾特拉⽅程組的⾏進波解之 L 型屏障最⼤值原理

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賴承志

*?2M@*?B? GB

指導教授 夏俊雄副教授

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中華民國 Ry8 年 e ⽉

CmM2- kyRe

感謝夏俊雄⽼師不厭其煩的指引我學習⽅向 培養獨⽴思考以及做研究的態度 除了積極地為我尋找磨練的機會之外 也包容任性的我 換了三個研究主題 並肯 定我的研究能⼒ 給了我很⼤的信⼼ 在就學期間 除了數學知識上的收穫 學到 更多的是做事以及做研究的態度與熱忱 並且讓我有機會能為數學科普的推廣盡⼀

份⼒ 另外 特別感謝陳俊全⽼師以及洪⽴昌學⾧栽培後進不遺餘⼒ 引領我進⼊

⽣物數學的世界 讓碩論的主題有了⽅向 最後 我要感謝建鑫 啟樺 世緯和春 華 除了互相切磋學習之外 也為研究⽣活增添了歡笑與活⼒

賴承志 kyReXye

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L 型屏障最⼤值原理為⼀種估計⼀維擴散@競爭洛特卡@佛爾特拉⽅程組的⾏進 波解之技術 這篇⽂章中 我們將 (9) 中考慮的雙物種之情況推廣到任意多物種 此外 我們將不再需要 為了得到更精細的估計 ⽽在 (9) 中所考慮的切線法之限 制條件

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i2K, ⎧

⎪⎨

⎪⎩

˜

ut= d1u˜yy+ ˜u(σ1− c¹¹u˜− c¹²˜v), y∈ R, t > 0,

˜

vt = d2˜vyy+ ˜v(σ2− c²¹u˜− c²²˜v), y ∈ R, t > 0,

URXyXRV

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EQHKQ;Q`Qp 2[miBQM

˜

ut = d1u˜yy+ ˜u(σ1− c11u), y˜ ∈ R, t > 0. URXyXkV

6Q` i?2 +b2 d1 = σ = c11= 1- EQKQH;Q`Qp- S2i`QpbFv M/ SBbFmMQp (8) T`Qp2/ i?i mM/2` i?2 BMBiBH +QM/BiBQM

˜

u(y, 0) =

⎧⎪

⎨

⎪⎩

1- 7Q` y < 0, 0- 7Q` y > 0,

R

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x = y− θKBMt- r?2`2 θKBM = 2 Bb i?2 KBMBKmK 2pQHpBM; bT22/ r?B+? rBHH #2 /Bb+mbb2/

KQ`2 BM i?2 TT2M/Bt ȜdX AM 7+i- i?2`2 2tBbib  7mM+iBQM ψ bm+? i?i

|˜u(y, t) − u(y − 2t − ψ(t))| → 0 b t → 0

mMB7Q`KHv BM x- M/ HBK

t→∞ψ^′(t) = 0X h?Bb KQiBpi2b mb iQ bim/v i`p2HBM; rp2 bQHmiBQMb Q7 URXyXRV,

(u(y, t), v(y, t)) = (u(x), v(x)), x = y− θt, URXyXjV r?2`2 θ biM/b 7Q` i?2 rp2 p2HQ+Biv Q7 i?2 i`p2HBM; rp2X am#biBimiBM; URXyXjV BMiQ URXyXRV- i?2 bvbi2K #2+QK2b  MQMHBM2` b2+QM/@Q`/2` Q`/BM`v /Bz2`2MiBH bvbi2K,

⎧⎪

⎨

⎪⎩

d₁u^′′+ θu^′+ u(σ₁− c11u− c12v) = 0, x∈ R, d₂v^′′+ θv^′ + v(σ₂− c21u− c22v) = 0, x∈ R.

URXyX9V

h?2`2 `2 7Qm` +?QB+2b Q7 i?2 `iB}+BH #QmM/`v +QM/BiBQMb (u, v)(−∞) = 2− M/

(u, v)(+∞) = 2+ 7Q` URXyX9V,

2

1 = (0, 0),

2

2 =

%σ1

c11

, 0

&

,

2

3 =

% 0, σ2

c22

&

M/ 24 =

% σ1c22− σ2c12

c11c22− c¹²c21

, σ2c11− σ1c21

c11c22− c¹²c21

&

,

r?B+? `2 i?2 bQHmiBQMb iQ i?2 H;2#`B+ 2[miBQMb,

⎧⎪

⎨

⎪⎩

u(σ1− c11u− c12v) = 0, x∈ R, v(σ2 − c21u− c22v) = 0, x∈ R.

 ivTB+H #QmM/`v +QM/BiBQM /Bb+mbb2/ BM (9) Bb i?2 (22,

2

3)@#QmM/`v +QM/BiBQMX h?i Bb-

(u, v)(−∞) =

%σ1

c₁₁, 0

&

, (u, v)(+∞) =

% 0, σ2

c₂₂

&

. URXyX8V

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⎧⎪

⎨

⎪⎩

˜

ut= ˜uyy+ ˜u(1− ˜u − a¹˜v), y∈ R, t > 0,

˜

vt= d˜vyy+ σ˜v(1− a²u˜− ˜v), y ∈ R, t > 0.

URXyXeV

r?2`2 ˜u(y, t) = ^c_σ¹¹₁u((_σ^d1

1)¹²y,_σ¹

1t), ˜v(y, t) = ^c_σ²²

2v((_σ^d1

1)¹²y,_σ¹

1t), a₁ = ^σ_σ²

1

c12

c22, a₂ =

σ1

σ2

c21

c11, d = ^d_d²₁ M/ σ = ^σ_σ²₁X h?2M i?2 +Q``2bTQM/BM; i`p2HBM; rp2 bQHmiBQM biBb@

}2b ⎧

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎩

u^′′+ θu^′+ u(1− u − a¹v) = 0, x∈ R, dv^′′+ θv^′+ σv(1− a²u− v) = 0, x ∈ R, (u, v)(−∞) = (1, 0), (u, v)(+∞) = (0, 1).

URXyXdV

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j

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kXR L"JS 7Q` k@bT2+B2b

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⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

d₁u^′′+ θu^′+ u(σ₁− c11u− c12v)≤ 0, x ∈ R, d₂v^′′+ θv^′+ v(σ₂− c21u− c22v)≤ 0, x ∈ R, (u, v)(−∞) = '

σ1

c11, 0(

, (u, v)(+∞) =' 0,_c^σ2

22

(.

UkXRXRV

amTTQb2 i?i

_c^σ₁₁¹ > _c^σ₂₁²

M/

_c^σ₂₂² > _c^σ₁₂¹

- i?2M 7Q` Mv α, β> 0 r2 ?p2 i?2 7QHHQrBM;

HQr2` #QmM/,

αu(x) + βv(x)≥ KBM )

ασ2

c21

, βσ1

c12

* KBM{d1, d2} Kt{d¹, d2}.

S`QQ7X 6Q` Mv ;Bp2M a, b > 0- r2 iF2 i?2 HBM2` +QK#BMiBQM Q7 i?2 }`bi irQ 2[miBQMb

BM UkXRXRV iQ ;2i

0≥ a[d1u^′′+ θu^′ + u(σ₁− c11u− c12v)] + b[d₂v^′′+ θv^′+ v(σ₂− c21u− c22v)]

= q^′′(x) + θp^′(x) + F (u(x), v(x)),

UkXRXkV

r?2`2 ⎧

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎩

q(x) = ad1u(x) + bd2v(x), p(x) = au(x) + bv(x),

F (u, v) = au(σ1 − c¹¹u− c¹²v) + bv(σ2 − c²¹u− c²²v).

6B`bi Q7 HH- r2 H2p2 i?2 }`bi irQ HBM2` i2`Kb q(x) M/ p(x)- M/ 7Q+mb QM i?2

9

MQMHBM2` i2`K F (u(x), v(x))X aBM+2 i?2 /2i2`KBMMi Q7 i?2 [m/`iB+ +m`p2 F (u, v) = 0 Bb

(ac12+ bc21)²− 4ac11bc22= a²c²₁₂+ 2abc12c21+ b²c²₂₁− 4abc11c22

> a²c²₁₂+ 2abc12c21+ b²c²₂₁− 4ab

^✚

σ

^✚

₁c₂₁

✚

σ

✚

2

✚

σ

✚

₂c₁₂

✚

σ

✚

1

= (ac12− bc²¹)² ≥ 0,

UkXRXjV

F (u, v) = 0 Bb  ?vT2`#QHX >2`2- r2 ?p2 mb2/ i?2 T`K2i2` bbmKTiBQMb _c^σ₁₁¹ > _c^σ₂₁²

M/ _c^σ₂₂² > _c^σ1

12X

6m`i?2`KQ`2- #v Q#b2`pBM; i?2 bB;Mb Q7 F (u, v) QM i?2 u@ M/ v@t2b M/ i?2 7+i i?i F (u, v) = 0 Tbb2b i?`Qm;? i?2 i?`22 TQBMib (0, 0)- (0,_c^σ₂₂²) M/ (_c^σ₁₁¹, 0)- r2 +QM+Hm/2 i?i (0,_c^σ₂₂²)M/ (_c^σ₁₁¹, 0)HB2 QM i?2 bK2 #`M+? r?BH2 (0, 0) HB2b QM i?2 Qi?2`

#`M+? Ub22 6B;m`2 RVX +imHHv- F (u, 0) = au(σ1− c11u) M/ F (0, v) = bv(σ2− c22v) BM/B+i2 i?i F (u, 0) > 0 7Q` 0 < u < <sub>c</sub><sup>σ</sup><sub>11</sub><sup>1</sup>- F (u, 0) < 0 7Q` u < 0 Q` u > <sub>c</sub><sup>σ</sup><sub>11</sub><sup>1</sup>- F (0, v) > 0 7Q` 0 < v < ^σ_c2²- M/ F (v, 0) < 0 7Q` v < 0 Q` v > _c^σ₂₂²X

σ

₁

c

₁₁

σ

₂

c

₂₂

+

+

−

−

−

− u v

6B;m`2 R, F (u, v) = 0 M/ i?2 bB;M Q7 F

q2 `2 `2/v iQ +QMbi`m+i Qm` L@#``B2`X 6B`biHv- #v i?2 T`K2i2` bbmKTiBQMb

σ1

c11 > _c^σ₂₁² M/ _c^σ₂₂² > _c^σ₁₂¹- i?2 BMi2`b2+iBQM Q7 i?2 irQ HBM2b σ1 − c11u− c12v = 0 M/

σ2 − c²¹u− c²²v = 0 Bb BM i?2 }`bi [m/`Mi- M/ Bi HbQ HB2b QM i?2 [m/`iB+ +m`p2 F (u, v) = au(σ1− c11u− c12v) + bv(σ2− c21u− c22v) = 0X h?2`27Q`2- i?2 HBM2 b2;K2Mi

8

#2ir22M (0,_c^σ₁₂¹) M/ (_c^σ₂₁², 0) HB2b mM/2`M2i? i?2 [m/`iB+ +m`p2 F (u, v) = 0X h?Bb HBM2 b2;K2Mi iQ;2i?2` rBi? i?2 u@tBb M/ v@tBb 7Q`K  `B;?i i`BM;H2 T HB2b 2MiB`2Hv

#2HQr i?2 +m`p2 F (u, v) = 0 BM i?2 }`bi [m/`Mi Ub22 6B;m`2 kVX h?mb F (u, v) > 0 7Q` HH (u, v) ∈ T X

σ

₂

c

₂₁

σ

₁

c

₁₁

σ

₁

c

₁₂

σ

2

c

22

T u

v

6B;m`2 k, #Hm2 HBM2b, σ1− c11u− c12v = 0 M/ σ2− c21u− c22v = 0c `2/ HBM2, i?2 HBM2 b2;K2Mi #2ir22M (0,_c^σ₁₂¹) M/ (_c^σ₂₁², 0)c M/ i?2 `B;?i i`BM;H2 T

X G2i mb /2MQi2

Qλ ={(u, v) | ad1u + bd₂v ≤ λ, u, v ≥ 0}

M/

Pη ={(u, v) | au + bv ≤ η, u, v ≥ 0}, +Q``2bTQM/BM; iQ i?2 HBM2` i2`Kb Q#iBM2/ BM UkXRXkVX

UBV h?2 }`bi HBM2 Q7 i?2 L@#``B2` Bb q = ad1u + bd2v = λ2- r?2`2 λ2 =bmT{λ | Q<sup>λ</sup> ⊂ T }. hQ #2 KQ`2 bT2+B}+- bBM+2 i?2 BMi2`+2Tib Q7 ad1u + bd2v = λ2 `2 '

λ2

ad1, 0(

M/ ' 0,_bd^λ2

2

(- M/ i?2 +QM/BiBQM Q^λ2 ⊂ T `2[mB`2b i?i _ad^λ2₁ ≤ _c^σ₂₁² M/ _bd^λ2₂ ≤ _c^σ₁₂¹-

?2M+2 λ2 =KBM+ ad1σ2

c21, bd2σ1

c12

,X

UBBV h?2 b2+QM/ HBM2 Q7 i?2 L@#``B2` Bb p = au + bv = η- r?2`2 η = bmT{η | P^η ⊂ Qλ2}. hQ #2 KQ`2 bT2+B}+- bBM+2 i?2 BMi2`+2Tib Q7 au + bv = η `2 -_η

a, 0. M/

e

-0,^η_b.- M/ i?2 +QM/BiBQM P^η ⊂ Qλ2 `2[mB`2b i?i ^η_a ≤ _ad^λ2₁ M/ ^η_b ≤ _bd^λ2₂- bQ η =KBM{^λ_d²₁,^λ_d²

2} = _Kt{d^λ2_1,d2}X

UBBBV h?2 i?B`/ HBM2 Q7 i?2 L@#``B2` Bb q = ad1u + bd2v = λ1- r?B+? Bb T`HH2H iQ i?2 }`bi HBM2- r?2`2 λ1 = bmT{λ | Q<sup>λ</sup> ⊂ P<sup>η</sup>}X hQ #2 KQ`2 bT2+B}+- bBM+2 i?2 BMi2`+2Tib Q7 ad1u + bd2v = λ1 `2 '

λ1

ad1, 0(

M/ ' 0,_bd^λ1₂(

- M/ i?2 +QM/BiBQM Q^λ1 ⊂ P^η `2[mB`2b i?i _ad^λ1₁ ≤ ^η_a, _bd^λ1

2 ≤ ^η_b- i?2`27Q`2 λ1 = KBM{d¹η, d2η} = λ2KBM{d1,d2}

Kt{d1,d2} =KBM+ ad1σ2

c21, bd2σ1

c12

, _KBM{d

1,d2} Kt{d1,d2}X

h?2 i?`22 HBM2b 2bi#HBb?2/ #Qp2 7Q`K i?2 L@#``B2` b 6B;m`2 jX Ai Bb 2bv iQ `2HBx2 i?i i?2 i2`K ǶL@#``B2`Ƕ +QK2b 7`QK i?2 `2b2K#HM+2 iQ i?2 1M;HBb? HT?#2i ǶLǶ- 2p2M i?2 b?T2 Q7 i?2 L@#``B2` Kv i?2 `2~2+iBQM Q7 i?2 +?`+i2` ǶLǶX

σ

₂

c

₂₁

σ

₁

c

₁₁

σ

1

c

₁₂

σ

2

c

22

u v

6B;m`2 j, i?2 L@#``B2`

LQr- r2 b?Qr λ1 Bb i?2 HQr2` #QmM/ b i?2 7QHHQrBM;,

*HBKX q(x) = ad

1u(x) + bd2v(x)≥ λ1

- 7Q` HH x ∈ RX

amTTQb2 +QMi``v- i?2M i?2`2 2tBbib z0 ∈ R bXiX q(z⁰) = ad1u(z0) + bd2v(z0) < λ1X aBM+2

λ1 =KBM) ad1

σ2

c21

, bd2

σ1

c12

* KBM{d1, d2} Kt{d¹, d2} ≤

⎧⎪

⎨

⎪⎩ ad1σ2

c21 < ad1σ1

c11

bd2σ1

c12 < bd2σ2

c22

d

M/

q(−∞) = ad¹σ1

c11

> λ1, q(+∞) = bd²σ2

c22

> λ1, UkXRX9V r2 Kv bbmK2 i?i q(z0) = KBM

x∈R q(x)X h?2`27Q`2- q^′(z₀) = 0X G2i z1M/ z2 #2 i?2 }`bi TQBMi i r?B+? i?2 bQHmiBQM (u(x), v(x)) BMi2`b2+ib i?2 }`bi HBM2 q = ad1u + bd2v = λ2

BM i?2 uv@THM2 r?2M x KQp2b 7`QK z0 iQr`/ −∞ M/ +∞- `2bT2+iBp2HvX h?i Bb-

z1 =BM7{z ∈ (−∞, z0)| q(x) = ad1u(x) + bd2v(x) < λ2, ∀x ∈ (z, z0)}

M/

z2 =bmT{z ∈ (z0, +∞) | q(x) = ad1u(x) + bd2v(x) < λ2, ∀x ∈ (z0, z)}.

>2M+2- q^′(z1) ≤ 0 M/ q^′(z2) ≥ 0X 6m`i?2`KQ`2- bBM+2 (u(z<sup>0</sup>), v(z0)) HB2b mM/2`M2i?

i?2 b2+QM/ HBM2 p = au+bv = η r?BH2 (u(z1), v(z₁))M/ (u(z2), v(z₂))HB2 #Qp2 r?B+?- p(z0) < η M/ p(z1), p(z2) > η Ub22 6B;m`2 9VX

σ

₁

c

₁₁

σ

₂

c

₂₂

−∞

+ ∞

z 2

z 1

z 0

ad 1 u + bd 2 v = q(z 0 ) (u(x), v(x))

u v

6B;m`2 9, z0, z1, z2 M/ bQHmiBQM +m`p2 (u(x), v(x))

AM //BiBQM- bBM+2 i?2 `+b {(u(x), v(x)) | x ∈ (z<sup>1</sup>, z0)} M/ {(u(x), v(v)) | x ∈ (z0, z2)} HB2 BM i?2 `B;?i i`BM;H2 T BM 6B;m`2 k- F (u(x), v(x)) > 0 7Q` HH x ∈ (z1, z0) Q` x ∈ (z⁰, z2)X

3

6Q` i?2 +b2 r?2`2 θ ≥ 0- BMi2;`iBM; UkXRXkV 7`QK z0 iQ z2 vB2H/b  +QMi`/B+iBQM,

q^′(z2) / 01 2

≥0

−

✟✟ ✟ ✟ ✯

0

q^′(z0) + θ(p(z2) / 01 2

>η

− p(z⁰) / 01 2

<η

) + 3 z2

z0

F (u(x), v(x))dx

/ 01 2

>0

≤ 0

→←

UkXRX8V

6Q` i?2 Qi?2` +b2 r?2`2 θ ≤ 0- r2 BMi2;`i2 UkXRXkV 7`QK z1 iQ z0 iQ Q#iBM  +QMi`@

/B+iBQM,

✟✟ ✟ ✟ ✯

0 q^′(z0)− q^′(z1)

/ 01 2

≤0

+θ(p(z0) / 01 2

<η

− p(z1) / 01 2

>η

) + 3 z0

z1

F (u(x), v(x))dx

/ 01 2

>0

≤ 0

→←

UkXRXeV

h?mb

d₁au(x) + d₂bv(x)≥ λ1 =KBM) ad₁σ2

c21

, bd₂σ1

c12

* KBM{d¹, d2} Kt{d1, d2}.

"v iFBM; a = _d^α₁, b = _d^β₂- r2 Q#iBM i?2 /2bB`2/ `2bmHi,

αu(x) + βv(x)≥ KBM )

ασ2

c21

, βσ1

c12

* KBM{d1, d2} Kt{d¹, d2}.

LQi2 i?i B7 d1 M/ d2 `2 2[mH- i?2 i?`22 HBM2b Q7 i?2 L@#``B2` BM i?2 T`QQ7 +QBM+B/2X >Qr2p2`- i?2 T`QQ7 #Qp2 biBHH rQ`FbX AM 7+i- i?2 }`bi HBM2 M/ b2+QM/ HBM2 Q7 i?2 L@#``B2` `2 mb2/ iQ /2H rBi? i?2 irQ HBM2` i2`Kb BM UkXRXkV- `2bT2+iBp2HvX

M/ i?2 i?B`/ HBM2 rQ`Fb BM i?2 T`QQ7 Q7 +QMi`/B+iBQM iQ b?Qr i?i q(z0) = KBM

x∈R q(x)X h?mb- B7 i?2 i?`22 HBM2b +QBM+B/2- i?2 irQ HBM2` i2`Kb #2+QK2 i?2 bK2- M/ r2 biBHH

?p2 q(z0) = KBM

x∈R q(x)X

aBKBH`Hv- #v +QMbi`m+iBM; M L@#``B2` #Qp2 i?2 [m/`iB+ +m`p2 F (u, v) = 0 Ub22 6B;m`2 8V- i?2 +Q``2bTQM/BM; mTT2` #QmM/ +M #2 Q#iBM2/ b 7QHHQrbX

N

h?2Q`2K kXk UlTT2` #QmM/VX G2i (u(x), v(x)) #2  MQMM2;iBp2 bQHmiBQM iQ

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

d1u^′′+ θu^′ + u(σ1− c11u− c12v)≥ 0, x ∈ R, d2v^′′+ θv^′+ v(σ2 − c21u− c22v)≥ 0, x ∈ R, (u, v)(−∞) ='

σ1

c11, 0(

, (u, v)(+∞) = ' 0,_c^σ₂₂²(

.

amTTQb2 i?i

_c^σ₁₁¹ > _c^σ2

21

M/

_c^σ₂₂² > _c^σ1

12

- i?2M 7Q` Mv α, β> 0 r2 ?p2 i?2 7QHHQrBM;

mTT2` #QmM/,

αu(x) + βv(x)≤ Kt )

ασ1

c₁₁, βσ2

c₂₂

* Kt{d1, d2} KBM{d1, d₂}.

S`QQ7X q2 QMHv b?Qr ?Qr iQ +QMbi`m+i M L@#``B2` #Qp2 F (u, v) = 0- M/ Q#iBM

λ1X 6Q` +?2+FBM; λ1 M mTT2` #QmM/ 7Q` αu + βv- i?2 `;mK2Mi Bb bK2 b r?B+? Q7 h?2Q`2K 2.1 M/ Bb ?2M+2 QKBii2/X

UBV h?2 }`bi HBM2 Q7 i?2 L@#``B2` Bb q = ad1u + bd₂v = λ₂X aBM+2 i?2 BMi2`+2Tib Q7 ad1u + bd2v = λ2 `2 '

λ2

ad1, 0(

M/ ' 0,_bd^λ2

2

(- M/ r2 `2[mB`2b _ad^λ2₁ ≥ _c^σ₁₁¹ M/

λ2

bd2 ≥ _c^σ₂₂²- ?2M+2 λ2 =Kt+ ad_1c^σ1

11, bd_2c^σ2

22

,X

UBBV h?2 b2+QM/ HBM2 Q7 i?2 L@#``B2` Bb p = au + bv = ηX aBM+2 i?2 BMi2`+2Tib Q7 au + bv = η `2 -_η

a, 0. M/ -0,^η_b.- M/ r2 `2[mB`2b ^η_a ≥ _ad^λ2₁ M/ ^η_b ≥ _bd^λ2₂- bQ η =Kt{^λ_d1²,^λ_d²₂} = _KBM{d^λ2_1,d2}X

UBBBV h?2 i?B`/ HBM2 Q7 i?2 L@#``B2` Bb q = ad1u + bd₂v = λ₁- r?B+? Bb T`HH2H iQ i?2 }`bi HBM2X aBM+2 i?2 BMi2`+2Tib Q7 ad1u + bd2v = λ1 `2'

λ1

ad1, 0(

M/' 0,_bd^λ1₂(

- M/

r2 `2[mB`2b _ad^λ1₁ ≥ ^η_a M/ _bd^λ1₂ ≥ ^η_b- i?2`27Q`2- λ1 =Kt{d1η, d₂η} = λ2Kt{d1,d2} KBM{d1,d2} = Kt+

ad1σ1

c11, bd2σ2

c22

,_Kt{d

1,d2} KBM{d1,d2}X

aBKBH`Hv- iFBM; a = <sub>d</sub><sup>α</sup><sub>1</sub> M/ b = <sub>d</sub><sup>β</sup><sub>2</sub>- r2 ?p2 i?2 /2bB`2/ mTT2` #QmM/X

Ry

−∞

+ ∞ z 2

z 1

z 0

u v

6B;m`2 8, L@#``B2` 7Q` mTT2` #QmM/

kXk :2M2`HBx2/ L"JS

_2+HH i?i i?2 T`K2i2` bbmKTiBQMb _c^σ₁₁¹ > _c^σ₂₁² M/ _c^σ₂₂² > _c^σ₁₂¹ BM h?2Q`2K 2.1 M/

h?2Q`2K 2.2 ?p2 #22M mb2/ iQ b?Qr i?i i?2 [m/`iB+ +m`p2 F (u, v) = 0 Bb  ?vT2`@

#QH BM UkXRXjV- i?i i?2 `B;?i i`BM;H2 T BM 6B;m`2 k HB2b #2HQr i?2 +m`p2 F(u, v) = 0

M/ i?i q(−∞) = ad¹_c^σ₁₁¹ > λ1, q(+∞) = bd²_c^σ₂₂² > λ1 BM UkXRX9VX

6B`bi Q7 HH- rBi?Qmi i?2b2 T`K2i2` bbmKTiBQMb- i?2 [m/`iB+ +m`p2 F (u, v) = 0 Kv #2  T`#QH Q`  2HHBTb2X 6Q`imMi2Hv- mbBM; i?2 TQbBiBpBiv Q7 i?2 +Q2{+B2Mib σi

M/ cij UB-D4R-kV- i?2 bB;Mb Q7 F (u, v) +M #2 b?QrM b 6B;m`2 eX

σ1 c11 σ2

c22

+

+

− −

−

− u v

σ1 c11 σ2

c22

+

+

− −

−

− u v

6B;m`2 e, H27i, T`#QHc `B;?i, 2HHBTb2

a2+QM/Hv- Bi Bb 2bv iQ b22 7`QK 6B;m`2 e i?i- 7Q` 2+? +b2- i?2`2 2tBbib  `B;?i i`BM;H2 T b BM 6B;m`2 k HB2b #2HQr i?2 +m`p2 F(u, v) = 0 BM i?2 }`bi [m/`Mi bQ i?i

RR

F (u, v) > 0 7Q` HH (u, v) ∈ T X h?2`27Q`2- L@#``B2`b +M HbQ #2 +QMbi`m+i2/ BM #Qi?

+b2bX

JQ`2Qp2`- 7Q` UkXRX9V- r2 QMHv M22/ iQ `2[mB`2 i?2 #QmM/`v +QM/BiBQMb 2₋ M/ 2+

HB2 #Qp2 i?2 }`bi HBM2 q = ad1u + bd2v = λ1X

*QMb2[m2MiHv- i?2 bbmKTiBQMb _c^σ₁₁¹ > _c^σ₂₁² M/ _c^σ₂₂² > _c^σ₁₂¹ +M #2 /`QTT2/X +im@

HHv- i?2 T`QQ7 HbQ rQ`Fb 7Q` F (u, v) = u<sup>m</sup>f (u, v) + v<sup>n</sup>g(u, v)- 7Q` +2`iBM ?vTQi?2b2b QM f(u, v)- g(u, v) M/ i?2 #QmM/`v +QM/BiBQMb 2+- 2₋- r?B+? rBHH #2 2tTHBM2/ BM h?2Q`2K 2.3 M/ h?2Q`2K 2.4 Ub22 6B;m`2 dVX

u v

u v

F (u, v) > 0

F (u, v) < 0

2 +

2 ₋

u v

6B;m`2 d, ;2M2`H +QM/BiBQMb

h?2Q`2K kXj U:2M2`HBx2/ HQr2` #QmM/VX G2i (u(x), v(x)) #2  MQMM2;iBp2 bQHmiBQM

iQ

⎧

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎩

d1u^′′+ θu^′ + u^mf (u, v)≤ 0, x ∈ R, d2v^′′+ θv^′+ vⁿg(u, v)≤ 0, x ∈ R, (u, v)(−∞) = 2−, (u, v)(+∞) = 2+.

amTTQb2 i?i i?2`2 2tBbib u > 0 M/ v > 0 bXiX f(u, v) > 0 M/ g(u, v) > 0- 7Q` HH

Rk

(u, v)∈ R M/ 2−,

2

+ ∈ [0, +∞)²\ R, r?2`2

R = )

(u, v)∈ [0, +∞)² 44 44u

u+ v v < 1

* .

h?2M 7Q` Mv α, β> 0 r2 ?p2 i?2 7QHHQrBM; HQr2` #QmM/,

αu(x) + βv(x)≥ KBM {αu, β v} KBM{d1, d2}

Kt{d¹, d2}.

h?2Q`2K kX9 U:2M2`HBx2/ mTT2` #QmM/VX G2i (u(x), v(x)) #2  MQMM2;iBp2 bQHmiBQM

iQ

⎧

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎩

d₁u^′′+ θu^′ + u^mf (u, v)≥ 0, x ∈ R, d2v^′′+ θv^′+ vⁿg(u, v)≥ 0, x ∈ R, (u, v)(−∞) = 2−, (u, v)(+∞) = 2+.

amTTQb2 i?i i?2`2 2tBbib u > 0 M/ v > 0 bXiX f(u, v) < 0 M/ g(u, v) < 0- 7Q` HH

(u, v)∈ R M/ 2−,

2

+ ∈ [0, +∞)²\ R, r?2`2

R =+

(u, v)∈ [0, +∞)²444u u+ v

v > 1, .

h?2M 7Q` Mv α, β> 0 r2 ?p2 i?2 7QHHQrBM; mTT2` #QmM/,

αu(x) + βv(x)≤ Kt {αu,β v}Kt{d¹, d2}

KBM{d1, d₂}.

kXj L"JS 7Q` JmHiB@bT2+B2b

"v `2TH+BM; HH ǶHBM2bǶ BM i?2 `;mK2Mi Q7 i?2 irQ@bT2+B2b +b2 /Bb+mbb2/ #Qp2 #v Ƕ?vT2`THM2bǶ- i?2 +Q``2bTQM/BM; `2bmHib 7Q` KmHiB@bT2+B2b +b2 `Bb2X

h?2Q`2K kX8 UJmHiB@bT2+B2b HQr2` #QmM/VX G2i (u

1(x),· · · , un(x))

#2  MQMM2;iBp2 bQHmiBQM iQ

⎧⎪

⎨

⎪⎩

diu^′′_i + θu^′_i+ u^m_i ⁱfi(u1,· · · , un)≤ 0, x ∈ R, i = 1, · · · , n, (u1,· · · , un)(−∞) = 2−, (u1,· · · , un)(+∞) = 2+.

UkXjXRV

Rj

bbmK2 i?i 7Q` 2+? i = 1, · · · , n- i?2`2 2tBbib u

i > 0

bXiX f

i(u1,· · · , un) > 0-

r?2M2p2` (u

1,· · · , uⁿ)∈ R- M/ 2−,

2

+ ∈ [0, +∞)ⁿ\ R, r?2`2

R = 5

(u1,· · · , un)∈ [0, +∞)ⁿ 44 44 4

6n i=1

u_i u_i < 1

7 .

h?2M 7Q` Mv α

i > 0- r2 ?p2 i?2 7QHHQrBM; HQr2` #QmM/,

6n i=1

αiui(x)≥

% KBM

i=1,··· ,nαiu_i

& KBM

i=1,··· ,ndi i=1,Kt··· ,ndi

, x∈ R.

S`QQ7X 6Q` Mv ;Bp2M a

1,· · · , aⁿ> 0- r2 iF2 i?2 HBM2` +QK#BMiBQM Q7 i?2 n 2[miBQMb BM UkXjXRV- r2 Q#iBM  bBM;H2 2[miBQM BMpQHpBM; p(x) M/ q(x)

q^′′(x) + p^′(x) + F (u1(x), u2(x), ..., un(x))≤ 0, x ∈ R, UkXjXkV

r?2`2 ⎧

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎩

q(x) = 8ⁿ

i=1

a_id_iu_i(x), p(x) =

8n i=1

aiui(x), F (u₁, u₂, ..., u_n) = 8ⁿ

i=1

α_iu^m_i ⁱf_i(u₁, u₂, ..., u_n).

h?2 +QMbi`m+iBQM Q7 i?2 L@#``B2` +QMbBbib Q7 /2i2`KBMBM; λ2- η- M/ λ1 bm+? i?i i?2 i?`22 ?vT2`THM2b 8ⁿ

i=1

ai diui = λ2- 8ⁿ

i=1

aiui = η M/ 8ⁿ

i=1

ai diui = λ1 2MDQv i?2 T`QT2`iv

Q^λ1 ⊂ P^η ⊂ Q^λ2 ⊂ R, UkXjXjV

r?2`2

Qλ = +

(u1, u2, ..., un)444 6n

i=1

ai diui ≤ λ, u1, u2, ..., un≥ 0,

; UkXjX9V

Pη = +

(u₁, u₂, ..., u_n)444 6n

i=1

a_iu_i ≤ η, u1, u₂, ..., u_n≥ 0,

. UkXjX8V

R9

q2 7QHHQr i?2 i?`22 bi2Tb #2HQr iQ +QMbi`m+i i?2 L@#``B2`, UBV h?2 }`bi ?vT2`THM2 Q7 i?2 L@#``B2` Bb q = 8ⁿ

i=1

aidiui = λ2- r?2`2 λ2 = bmT{λ | Qλ ⊂ R}X hQ #2 KQ`2 bT2+B}+- bBM+2 i?2 BMi2`+2Tib Q7 8ⁿ

i=1

aidiui = λ2

`2 ( λa1d<sup>2</sup>1, 0, ..., 0)- (0, λa2d<sup>2</sup>2, 0, ..., 0)-XXX- M/ (0, 0, ..., 0, λand<sup>2</sup>n)- M/ i?2 +QM/BiBQM Qλ2 ⊂ R `2[mB`2b i?i λ₂

aidi ≤ ui 7Q` i = 1, 2, ..., n- ?2M+2 λ2 = KBM

i=1,··· ,naidiu_iX UBBV h?2 b2+QM/ ?vT2`THM2 Q7 i?2 L@#``B2` Bb p = 8ⁿ

i=1

aiui = η- r?2`2 η = bmT{η | P<sup>η</sup> ⊂ Qλ2}. hQ #2 KQ`2 bT2+B}+- bBM+2 i?2 BMi2`+2Tib Q7 8ⁿ

i=1

aiui = η `2( ηa1, 0, ..., 0)- (0, η

a₂, 0, ..., 0)-XXX- M/ (0, 0, ..., 0, ηa_n)- M/ i?2 +QM/BiBQM P^η ⊂ Q^λ2 `2[mB`2b i?i aηi ≤ λ₂

aidi 7Q` i = 1, 2, ..., n- bQ η = KBM

i=1,··· ,n λ2

di = _Kt^λ2

i=1,··· ,ndiX UBBBV h?2 i?B`/ ?vT2`THM2 Q7 i?2 L@#``B2` Bb q = 8ⁿ

i=1

aidiui = λ1- r?B+? Bb T`HH2H iQ i?2 }`bi HBM2- r?2`2 λ1 = bmT{λ | Q<sup>λ</sup> ⊂ P<sup>η</sup>}X hQ #2 KQ`2 bT2+B}+- bBM+2 i?2 BMi2`+2Tib Q7 8ⁿ

i=1

aidiui = λ1 `2 ( λa1¹d1, 0, ..., 0)- (0, λa2¹d2, 0, ..., 0)-XXX- M/

(0, 0, ..., 0, λan¹dn)- M/ i?2 +QM/BiBQM Q^λ1 ⊂ P^η `2[mB`2b i?i λai¹di ≤ η ai 7Q`

i = 1, 2, ..., n- i?2`27Q`2 λ1 = KBM

i=1,··· ,ndiη = λ2

i=1,KBM··· ,ndi

i=1,Kt··· ,ndi = KBM

i=1,··· ,naidiu_i^i=1,_Kt^{KBM··· ,ndi}

i=1,··· ,ndiX h?2 i?`22 ?vT2`THM2b 8ⁿ

i=1

αi diui = λ2- 8ⁿ

i=1

αiui = η M/ 8ⁿ

i=1

αi diui = λ1 +QMbi`m+i2/

#Qp2 7Q`K i?2 L@#``B2`X

LQr- r2 b?Qr λ1 Bb i?2 HQr2` #QmM/ b i?2 7QHHQrBM;,

*HBKX q(x) =

8ⁿ

i=1

aidiui(x)≥ λ¹

- 7Q` HH x ∈ RX

amTTQb2 i?i- +QMi``v iQ Qm` +HBK- i?2`2 2tBbib z0 ∈ R bm+? i?i q(z0) = 8n

i=1

aidiui(z0) < λ1X 6`QK 2₋,

2

+ ∈ [0, +∞)ⁿ \ R M/ Q^λ1 ⊂ R- r2 FMQr i?i q(±∞) > λ1X aQ r2 Kv bbmK2 KBM

x∈R q(x) = q(z0)X h?2`27Q`2- q^′(z0) = 0X G2i z1 M/ z2 #2 i?2 }`bi TQBMi i r?B+? i?2 bQHmiBQM (u1(x),· · · , u<sup>n</sup>(x)) BMi2`b2+ib }`bi

?vT2`THM2 q = 8ⁿ

i=1

aidiui = λ2 BM i?2 u1· · · un@bT+2 r?2M x KQp2b 7`QK z0 iQr`/

R8

−∞ M/ +∞- `2bT2+iBp2HvX h?i Bb-

z1 =BM7{z ∈ (−∞, z⁰)| q(x) = 6n

i=1

aidiui(x) < λ2, ∀x ∈ (z, z⁰)}

M/

z2 =bmT{z ∈ (z⁰, +∞) | q(x) = 6n

i=1

aidiui(x) < λ2, ∀x ∈ (z⁰, z)}.

>2M+2- q^′(z1) ≤ 0 M/ q^′(z2) ≥ 0X 6m`i?2`KQ`2- bBM+2 (u¹(z0),· · · , uⁿ(z0)) HB2b mM@

/2`M2i? i?2 b2+QM/ ?vT2`THM2 p = 8ⁿ

i=1

aiui = η r?BH2 (u1(z1),· · · , un(z1)) M/

(u1(z2),· · · , uⁿ(z2))HB2 #Qp2 r?B+?- p(z0) < ηM/ p(z1)- p(z2) > ηX AM //BiBQM- bBM+2 i?2 `+b {(u1(x),· · · , un(x)) | x ∈ (z1, z0)} M/ {(u1(x),· · · , un(x)) | x ∈ (z0, z2)} HB2 BM Q<sup>λ</sup>2 ⊂ R- F (u<sup>1</sup>(x),· · · , u<sup>n</sup>(x)) > 0 7Q` HH x ∈ (z¹, z0) Q` x ∈ (z⁰, z2)X

6Q` i?2 +b2 r?2`2 θ ≥ 0- BMi2;`iBM; UkXjXkV 7`QK z⁰ iQ z2 vB2H/b  +QMi`/B+iBQM,

q^′(z2) / 01 2

≥0

−

✟✟ ✟ ✟ ✯

0

q^′(z0) + θ(p(z2) / 01 2

>η

− p(z0) / 01 2

<η

) + 3 z2

z0

F (u1(x),· · · , un(x))dx

/ 01 2

>0

≤ 0

→←

6Q` i?2 Qi?2` +b2 r?2`2 θ ≤ 0- r2 BMi2;`i2 UkXjXkV 7`QK z<sup>1</sup> iQ z0 iQ Q#iBM  +QMi`@

/B+iBQM,

✟✟ ✟ ✟ ✯

0 q^′(z0)− q^′(z1)

/ 01 2

≤0

+θ(p(z0) / 01 2

<η

− p(z1) / 01 2

>η

) + 3 z0

z1

F (u1(x),· · · , un(x))dx

/ 01 2

>0

≤ 0

→←

h?mb 6n

i=1

a_id_iu_i(x)≥ λ1 =

% KBM

i=1,··· ,na_id_iu_i

& KBM

i=1,··· ,nd_i

i=1,Kt··· ,ndi

.

"v iFBM; ai = ^α_diⁱ- r2 Q#iBM i?2 /2bB`2/ `2bmHi,

6n i=1

α_iu_i(x)≥

% KBM

i=1,··· ,nα_iu_i

& KBM

i=1,··· ,nd_i

i=1,Kt··· ,ndi

.

Re

LQi2 i?i B7 i?2 /BzmbBQM `i2b diǶb `2 HH 2[mH- i?2M i?2 i?`22 ?vT2`THM2b Q7 i?2 L@#``B2` +QBM+B/2X L2p2`i?2H2bb- i?2 T`QQ7 +M biBHH #2 ++QKTHBb?2/ b `2K`F2/

7i2` i?2 T`QQ7 Q7 h?2Q`2K 2.1X

h?2Q`2K kXe UJmHiB@bT2+B2b mTT2` #QmM/VX G2i (u

1(x),· · · , un(x))

#2  MQMM2;iBp2 bQHmiBQM iQ

⎧⎪

⎨

⎪⎩

diu^′′_i + θu^′_i+ u^m_i ⁱfi(u1,· · · , un)≥ 0, x ∈ R, i = 1, · · · , n, (u1,· · · , un)(−∞) = 2−, (u1,· · · , un)(+∞) = 2+.

bbmK2 i?i 7Q` i = 1, · · · , n- i?2`2 2tBbib u

i > 0

bXiX f

i(u₁,· · · , un) < 0- r?2M2p2`

(u1,· · · , un)∈ R- M/ 2−,

2

+∈ [0, +∞)ⁿ\ R, r?2`2

R = 5

(u1,· · · , uⁿ)∈ [0, +∞)ⁿ 44 44 4

6n i=1

ui

u_i > 1 7

.

h?2M 7Q` Mv α

i > 0- r2 ?p2 i?2 7QHHQrBM; mTT2` #QmM/,

6n i=1

α_iu_i(x)≤

% Kt

i=1,··· ,nα_iu_i

& Kt

i=1,··· ,ndi i=1,KBM··· ,ndi

.

Hi?Qm;? i?2 HQr2` M/ mTT2` #QmM/b +?B2p2/ #Qp2 `2 mb2/ 7Q` i?2 i`p2HBM;

rp2 bQHmiBQMb Q7 i?2 GQiF@oQHi2`` bvbi2Kb- bv URXyXRV- i?2v +imHHv +M #2 TTHB2/

iQ i?2 bi2/v bii2 bQHmiBQMb 7Q` URXyXRV,

⎧⎪

⎨

⎪⎩

d1u^′′+ u(σ1− c¹¹u− c¹²v) = 0, x∈ R, d2v^′′+ v(σ2− c²¹u− c²²v) = 0, x∈ R.

UkXjXeV

LK2Hv- θ = 0 BM i?2 i`p2HBM; rp2 p2`bBQM URXyX9VX AM 7+i- θ = 0 BMpHB/i2b i?2

z2+i Q7 i?2 HBM2` i2`K p BM UkXRX8V M/ UkXRXeVX h?2`27Q`2- i?2 }`bi HBM2 Q7 i?2 L@

#``B2` Bb i?2 QMHv HBM2 mb2/ BM i?2 T`QQ7- M/ i?2 2biBKi2b Q7 αu + βv BM i?2 bi2/v bii2b `2 #2ii2` i?M r?B+? BM i?2 i`p2HBM; rp2bX h?i Bb iQ bv-

KBM) ασ₂

c21

, βσ₁ c12

*

≤ αu + βv ≤ Kt )

ασ₁ c11

, βσ₂ c22

* , Rd

r?B+? Bb BM/2T2M/2Mi Q7 d1 M/ d2- 7Q` i?2 bQHmiBQMb iQ i?2 bi2/v bii2 bvbi2K UkXjXeVX

j TTHB+iBQM, LQM2tBbi2M+2 _2bmHib

h?2 2tBbi2M+2 Q7 i`p2HBM; rp2 bQHmiBQMb Q7 i?`22@bT2+B2b /BzmbBp2 +QKT2iBiBp2 GQiF@oQHi2`` bvbi2Kb Bb +?B2p2/ BM (k)X *?2M- >mM;- JBKm` M/ l2vK K/2

M Mbix i?i

(u(x), v(x), w(x)) = (k₁(1 +iM? x), k2(1 +iM? x)², k₃(1 +iM?²x))

M/ p2`B}2/ Bi M 2t+i bQHmiBQM mM/2` +2`iBM T`K2i2`b M/ 7Q` bmBi#H2 k1- k2 M/

k3X

PM i?2 Qi?2` ?M/- i?2 MQM2tBbi2M+2 Q7 i`p2HBM; rp2 bQHmiBQMb Q7 i?`22@bT2+B2b /BzmbBp2 +QKT2iBiBp2 GQiF@oQHi2`` bvbi2Kb +M #2 +?B2p2/ rBi? i?2 B/ Q7 h?2Q@

`2K 2.1X

h?2Q`2K jXR ULQM2tBbi2M+2 Q7 j@bT2+B2b rp2VX amTTQb2 i?i

(>R) ˜σ

1 := σ1− c¹³_c^σ₃₃³ > 0, ˜σ2 := σ2− c²³_c^σ₃₃³ > 0,

(>k) KBM{

^c31_c21^σ˜2,^c32_c12^σ˜1}_Kt{d^KBM{d1₁^,d_,d²₂^}_} ≥ σ3.

h?2M i?2 i?`22@bT2+B2b GQiF@oQHi2`` bvbi2K

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

d₁u^′′+ θu^′ + u(σ₁− c11u− c12v− c13w) = 0, x∈ R, d₂v^′′+ θv^′+ v(σ₂ − c21u− c22v − c23w) = 0, x ∈ R, d₃w^′′+ θw^′+ w(σ₃− c31u− c32v− c33w) = 0, x ∈ R, (u, v, w)(−∞) ='

σ1

c11, 0, 0(

, (u, v, w)(+∞) =' 0,_c^σ2

22, 0(

UjXyXRV

?b MQ TQbBiBp2 bQHmiBQM (u(x), v(x), w(x))X

R3

S`QQ7X amTTQb2 +QMi``v- i?2M i?2`2 2tBbib  bQHmiBQM (u(x), v(x), w(x))- r?2`2 u(x)-

v(x)- M/ w(x) > 0- 7Q` HH x ∈ RX ++Q`/BM; iQ i?2 #QmM/`v +QM/BiBQM rU±∞V4y- i?2`2 Kmbi #2  x0 ∈ R bXiX w(x0) =Kt wX i i?Bb TQBMi- i?2 i?B`/ 2[mHBiv Q7 i?2 bvbi2K UjXyXRV #2+QK2b

d3

/012>0

w^′′(x0) / 01 2

≤0

+θ

✘✘✘ ✘ ✿

0 w^′(x0) + w(x0)

/ 01 2

>0

(σ3− c31u(x0)− c32v(x0)− c33w(x0)) = 0.

h?Bb b?Qrb i?i

σ3− c31u(x0)− c32v(x0)− c33w(x0)≥ 0. UjXyXkV

>2M+2- w(x) ≤ w(x0)≤ _c¹₃₃(σ3− c31u(x0)− c32v(x0)) < _c^σ₃₃³, ∀x ∈ RX am#biBimiBM; i?2 mTT2` #QmM/ 7Q` w BMiQ i?2 }`bi irQ 2[miBQMb BM UjXyXRV- i?2 i?`22@bT2+B2b +b2 rBHH

#2 `2/m+2/ BMiQ i?2 irQ@bT2+B2b +b2,

⎧⎪

⎨

⎪⎩

d1u^′′+ θu^′+ u(σ1− c¹¹u− c¹²v− c¹³_c^σ₃₃³) < 0, d2v^′′+ θv^′ + v(σ2− c²¹u− c²²v− c²³_c^σ₃₃³) < 0,

Q` ⎧

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎩

d1u^′′+ θu^′+ u((σ1− c¹³σ3

c₃₃)

/ 01 2

˜ σ1

−c¹¹u− c¹²v) < 0,

d2v^′′+ θv^′+ v((σ2− c²³σ3

c33

)

/ 01 2

˜ σ2

−c²¹u− c²²v) < 0.

TTHv h?2Q`2K 2.1 rBi? α = c31, β = c32 M/ 2tTHQBi i?2 ?vTQi?2bBb (>k) iQ ;2i i?2 HQr2` #QmM/,

c31u + c32v ≥ KBM

)c₃₁σ˜₂ c21

,c₃₂σ˜₁ c12

* KBM{d1, d₂} Kt{d1, d2} ≥ σ3.

"mi 7`QK UjXyXkV-

w(x0)≤ 1 c33

(σ3− c³¹u(x0)− c³²v(x0))≤ 0,

 +QMi`/B+iBQMX

RN

LQi2 i?i i?2 bi2Tb Q7 i?2 T`QQ7 7Q` i?2 MQM2tBbi2M+2 Q7 i?`22@bT2+B2b rp2b `2 i?i- }`bi- `2/m+BM; i?2 i?`22 2[miBQMb iQ irQ BM2[mHBiB2b M/- b2+QM/- TTHvBM; i?2 HQr2` #QmM/ Q#iBM2/ 7`QK i?2 L@#``B2` KtBKmK T`BM+BTH2 iQ `2+?  +QMi`/B+iBQMX

*QMb2[m2MiHv- QM+2 i?2 L@#``B2` KtBKmK T`BM+BTH2 7Q` KmHiB@bT2+B2b h?2Q`2K 2.5

?b #22M 2bi#HBb?2/- i?2 MQM2tBbi2M+2 Q7 KmHiB@bT2+B2b rp2b 7QHHQrb,

h?2Q`2K jXk ULQM2tBbi2M+2 `2bmHi 7Q` KmHiB@bT2+B2bVX amTTQb2 i?i

⎧⎪

⎪⎨

⎪⎪

⎩

˜

σi := σi− cin σn

cnn > 0, i = 1,· · · , n − 1,

i=1,KBM··· ,n

)

cni KBM

j=1,··· ,n

˜ σi

cji

* _KBM

i=1,··· ,ndi i=1,··· ,nKt di ≥ σn.

h?2M i?2 n@bT2+B2b GQiF@oQHi2`` bvbi2K

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

d_iu^′′_i + θu^′_i+ u_i 9

σ_i−8ⁿ

j=1

c_iju_j :

= 0 , x∈ R, i = 1, · · · , n, (u1,· · · , uⁿ)(−∞) = '

σ1

c11, 0,· · · , 0( , (u1,· · · , uⁿ)(+∞) ='

0,_c^σ2

22, 0,· · · , 0(

?b MQ TQbBiBp2 bQHmiBQM (u

1(x),· · · , un(x))X

6Q` MQi?2` TTHB+iBQM Q7 i?2 L@#``B2` KtBKmK T`BM+BTH2- i?2 `2/2`b `2 `2@

72``2/ iQ (j) 7Q` i?2 2tBbi2M+2 Q7 i?`22@bT2+B2b rp2b mM/2`  /Bz2`2Mi #QmM/`v +QM@

/BiBQMX >mM; }`bi `2/m+2/ i?2 i?`22 2[miBQMb iQ  bBM;H2 2[miBQM M/ 2KTHQv2/

i?2 L@#``B2` KtBKmK T`BM+BTH2 iQ +QMbi`m+i  bm#bQHmiBQM- i?2M i?2 K2i?Q/ Q7 bmT2`bQHmiBQM@bm#bQHmiBQM ;m`Mi22  bQHmiBQMX

9 AKT`Qp2/ hM;2Mi GBM2 J2i?Q/

AM i?Bb b2+iBQM-  `2}M2/ HQr2` #QmM/ 7Q` αu + βv rBHH #2 /2`Bp2/X 6Q` +QKTmi@

iBQMH +QMp2MB2M+2- r2 +QMbB/2` i?2 b+H2/ bvbi2K URXyXdV rBi? i?2 #Bbi#H2 +QM/BiBQM, a1- a2 > 1 M/ TTHv i?2 BKT`Qp2/ iM;2Mi HBM2 K2i?Q/X

ky

AM (9)- Bi Bb b?QrM i?i- mM/2` +2`iBM `2bi`B+iBQMb QM i?2 T`K2i2`b- i?2 HQr2`

#QmM/ BM h?2Q`2K 2.1 +M #2 BKT`Qp2/ #v K2Mb Q7 i?2 iM;2Mi HBM2 K2i?Q/X >Qr@

2p2`- i?2 `2bi`B+iBQMb `2 `2/mM/Mi- bBM+2 i?2 B/2 Bb iQ 2MH`;2 λ2bQ i?i i?2 L@#``B2`

biBHH HB2b mM/2`M2i? i?2 [m/`iB+ +m`p2,

F (u, v) = au(1− u − a1v) + bσv(1− a2u− v) = 0.

hQ #2 KQ`2 bT2+B}+,

Qλ1 ⊂ Pη ⊂ Qλ2 ⊂ R, U9XyXRV

r?2`2

P^η = +

(u, v)444 au + bv ≤ η, u, v ≥ 0,

; U9XyXkV

Q^λ = +

(u, v)444 au + dbv ≤ λ, u, v ≥ 0,

; U9XyXjV

R = +

(u, v)444 F (u, v) ≥ 0, u, v ≥ 0,

. U9XyX9V

AM 7+i- λ2 +M #2 ;Bp2M #v

λ2 =bmT{λ | Qλ ⊂ R}. U9XyX8V

_2TH+BM; i?2 }`bi bi2T 7Q` /2i2`KBMBM; λ2 BM h?2Q`2K 2.1 #v U9XyX8V-  bi`QM;2` HQr2`

#QmM/ i?M i?2 QM2 ;Bp2M #v h?2Q`2K 2.1 +M #2 7QmM/X AM Qi?2` rQ`/b- i?2 2biBKi2 +M #2 `2}M2/ rBi?Qmi Mv 7m`i?2` `2bi`B+iBQM QM i?2 T`K2i2`bX

hQ +H+mHi2 λ2- r2 }`bi bQHp2 v b  7mM+iBQM Q7 u BM i?2 ?vT2`#QH

F (u, v) = bkv² + (aa1u + bσ(a2u− 1))v + au(u − 1) = 0,

M/ +?QQb2 i?2 #`M+? r?B+? /Q2b MQi Tbb i?`Qm;? i?2 Q`B;BMX h?i Bb-

v(u) = −(aa1u + bσ(a2u− 1)) +;

(aa1u + bσ(a2u− 1))²− 4abσu(u − 1)

2bσ .

kR

aQ i?2 iM;2Mi iQ i?2 +m`p2 Bb

dv du(u) =

−(aa1+ bσa₂) + ^(aa1√^{u+bσ(a2u−1))(aa1+bσa2)}−2abσ(2u−1) (aa1u+bσ(a2u−1))²−4abσu(u−1)

2bσ .

6Q` ;Bp2M a- b M/ d- i?2 bHQT2 Q7 i?2 HBM2 au + dbv = λ2 Bb /2i2`KBM2/ #v ^−a_dbX h?2 bmT`2KmK 2tT`2bbBQM U9XyX8V b?Qrb i?i i?2 HBM2 au + dbv = λ2 b?QmH/ iM;2Mi iQ i?2

?vT2`#QH F = 0X >Qr2p2`- bBM+2 r2 `2 rQ`FBM; BM i?2 }`bi [m/`Mi- i?2`2 `2 irQ +`BiB+H iM;2Mib, <sup>dv</sup><sub>du</sub>(0) = <sup>−a(a1−1)−bσa</sup><sub>bσ</sub> <sup>2</sup> M/ <sup>dv</sup><sub>du</sub>(1) = <sub>aa1+bσ(a</sub><sup>−a</sup><sub>2−1)</sub>X "2+mb2 i?2 #`M+?

Q7 i?2 ?vT2`#QH r2 ?p2 +?Qb2M Bb +QMp2t Ub22 6B;m`2 RV- i?2 iM;2Mi iQ i?2 +m`p2 v(u) Bb BM+`2bBM;X h?mb- i?2`2 `2 i?`22 +b2b iQ #2 +QMbB/2`2/,

UBV ^−a_db < ^{−a(a1−1)−bσa}_bσ ²,

AM i?Bb +b2- i?2 }`bi HBM2 Q7 L@#``B2` au+dbv = λ2Tbb2b i?`Qm;? i?2 #QmM/`v (0, 1)- bQ λ2 Bb /2i2`KBM2/ b

λ2 = a· 0 + db · 1 = db.

1 1

z2

z1

u v

6B;m`2 3, L@#``B2` 7Q` +b2 UBV

LQi2 i?i z2 Kv #2 +∞ BM i?2 T`QQ7 Q7 h?2Q`2K 2.1 BM i?Bb +b2X AM 7+i-

x→+∞HBK q^′(x) ≥ 0- HBK_x

→+∞p(x) > η M/ <+∞

z0 F (u(x), v(x))dx > 0 biBHH `2+?  +QMi`/B+iBQM b UkXRX8VX

kk

UBBV ^−a_db > _aa1+bσ(a^−a₂₋₁₎,

AM i?Bb +b2- i?2 }`bi HBM2 Q7 L@#``B2` au+dbv = λ2Tbb2b i?`Qm;? i?2 #QmM/`v (1, 0)- bQ λ2 Bb /2i2`KBM2/ b

λ2 = a· 1 + db · 0 = a.

1 1

z1

z2

u v

6B;m`2 N, L@#``B2` 7Q` +b2 UBBV

LQi2 i?i z1 Kv #2 −∞ BM i?2 T`QQ7 Q7 h?2Q`2K 2.1 BM i?Bb +b2X AM 7+i-

x→−∞HBK q^′(x) ≤ 0- HBK

x→−∞p(x) > η M/ <z0

−∞F (u(x), v(x))dx > 0 biBHH `2+?  +QM@

i`/B+iBQM b UkXRX8VX

UBBBV ^{−a(a1−1)−bσa}_bσ ² < ^−a_db < _aa1+bσ(a^−a₂₋₁₎,

AM i?Bb +b2- i?2 }`bi HBM2 Q7 L@#``B2` au + dbv = λ2 Bb iM;2Mi iQ i?2 +m`p2 v(u)X

kj

1 1

z₁ z2

u v

6B;m`2 Ry, L@#``B2` 7Q` +b2 UBBBV

h?2`27Q`2-

dv du(u) =

−(aa1+ bσa2) + ^(aa1√^{u+bσ(a2u−1))(aa1+bσa2})−2abσ(2u−1) (aa1u+bσ(a2u−1))²−4abσu(u−1)

2bσ = −a

db, Q`

[(X²− 4abσ)u + (−bσX + 2abσ)]² (Xu− bσ)²− 4abσu(u − 1) =

%

X− 2aσ d

&2

, r?2`2 X = aa1+ bσa2X h?2M Bi #2+QK2b

Au²+ Bu + C

Du²+ Eu + F = G, U9XyXeV

r?2`2

A =-

X² − 4abσ.2

,

B = 2(X²− 4abσ)(−bσX + 2abσ), C = (−bσX + 2abσ)²,

D = X²− 4abσ, E =−2bσX + 4abσ,

F = b²σ²,

k9

G =

%

X− 2aσ d

&2

. q`Bi2 U9XyXeV b

(A− DG)u²+ (B− EG)u + (C − F G) = 0,

r2 ?p2

u0 = −(B − EG) ±;

(B− EG)²− 4(A − DG)(C − F G)

2(A− DG) .

>2M+2-

λ2 =au0+ dbv(u0)

=au0+ db−(aa1u0+ bσ(a2u0− 1)) +;

(aa1u + bσ(a2u0− 1))²− 4abσu0(u0− 1)

2bσ ,

r?2`2

u0 = −(B − EG) ±;

(B− EG)²− 4(A − DG)(C − F G) 2(A− DG)

M/ i?2 #`M+? Bb +?Qb2M bXiX

0 < u0 < 1

M/

dv

du(u0) =

−X + √^(Xu0−bσ)X−2abk(2u0−1) (Xu0−bσ)²−4abσu0(u0−1)

2bσ = −a

db.

LQi2 i?i i?2 #`M+? Q7 u0 +M MQi #2 /2i2`KBM2/ mMH2bb i?2 +Q2{+B2Mib `2 ;Bp2MX

+imHHv- #v mbBM; i?2 +QKTmi2` T`Q;`K JhG"- r2 +M b?Qr i?i 7Q` a = 12- b = 2- a1 = 5- a2 = 3- d = 2 M/ σ = 6- u0 b?QmH/ #2 ^−(B−EG)+√

(B−EG)²−4(A−DG)(C−F G)

2(A−DG) c

r?BH2 7Q` a = 12- b = 1- a1 = 12- a2 = 2- d = 1 M/ σ = 12- u0 b?QmH/ #2

−(B−EG)−√

(B−EG)²−4(A−DG)(C−F G)

2(A−DG) X

_2+HH i?i BM i?2 T`QQ7 Q7 h?2Q`2K 2.1 r2 iQQF a = _d^α₁ M/ b = _d^β₂c r?BH2 BM i?Bb

k8

`2b+H2/ +b2- a = α M/ b = <sup>β</sup><sub>d</sub>X AM +QM+HmbBQM- r2 ?p2 i?2 7QHHQrBM; `2}M2/ HQr2`

#QmM/,

h?2Q`2K 9XR U_2}M2/ 2biBKi2VX G2i (u(x), v(x)) #2  MQMM2;iBp2 bQHmiBQM iQ

URXyXdV- BX2X ⎧

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎩

u^′′+ θu^′+ u(1− u − a¹v) = 0, x∈ R, dv^′′+ θv^′ + σv(1− a²u− v) = 0, x ∈ R, (u, v)(−∞) = (1, 0), (u, v)(+∞) = (0, 1),

r?2`2 a

1

- a

2 > 1X h?2M 7Q` Mv α, β> 0 r2 ?p2 i?2 7QHHQrBM; HQr2` #QmM/,

αu(x) + βv(x)≥ λ1KBM{1, d}

Kt{1, d},

r?2`2

λ1 =

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

β

- B7

^−α_β < ^{−α(a1−1)−}β ^βdσa2

dσ ,

α

- B7

^−α_β > ^−α

αa1+^β_dσ(a2−1), αu0+ β^−(αa1u0+

β

dσ(a2u0−1))+√

(αa1u0+^β_dσ(a2u0−1))²−4^αβ_d σu0(u0−1)

2^β_dσ

- Qi?2`rBb2-

BM r?B+? u

0 = ^{−(B−EG)±}

√(B−EG)²−4(A−DG)(C−F G)

2(A−DG)

- r?2`2 A =

-

X²− 4^αβ_d σ.2

, B = 2(X²−4^αβ_d σ)(−^β_dσX+2^αβ_d σ), C =-

−^β_dσX + 2^αβ_d σ.2

, D = X²−4^αβ_d σ, E =−2^β_dσX+

4^αβ_d σ, F = ^β_d2²σ²

M/ G =

-

X−^2ασ_d .2

BM r?B+? X = αa

1+^β_dσa2

X h?2 #`M+? Bb +?Qb2M bXiX 0 < u

0 < 1

M/

%

−X +√^{(Xu0−βdσ)X−2αβdσ(2u0−1)}

(Xu0−^β_dσ)²−4^αβ_d σu0(u0−1)

&

/(2^β_dσ) = ^−α_β

X

8 1tKTH2b

1t+i bQHmiBQMb iQ irQ@ M/ i?`22@bT2+B2b GQiF@oQHi2`` bvbi2Kb `2 T`QTQb2/

BM (3) M/ (k)- `2bT2+iBp2HvX "Qi? 2tKTH2b rBHH #2 T2`7Q`K2/ BM i?Bb b2+iBQMX 6Q`

i?2 i?`22@bT2+B2b 2t+i bQHmiBQM- r2 rBHH +?2+F i?i i?2 mTT2` M/ HQr2` #QmM/b BM h?2Q`2K 2.5 M/ h?2Q`2K 2.6 `2 pHB/X 6Q` i?2 irQ@bT2+B2b 2t+i bQHmiBQM- r2 }`bi +QKTmi2 i?2 HQr2` #QmM/ pB i?2 BKT`Qp2/ iM;2Mi HBM2 K2i?Q/ h?2Q`2K 4.1 M/

ke

+?2+F i?2 HQr2` #QmM/ pHB/X h?2M +QKT`2 Bi rBi? i?2 HQr2` #QmM/ Q#iBM2/ 7`QK i?2 Q`B;BMH L@#``B2` KtBKmK T`BM+BTH2 h?2Q`2K 2.1X

8XR M 1tKTH2 Q7 L"JS 7Q` j@bT2+B2b

"v i?2 Mbix i?i ⎧

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎩

u(x) = k1(1− iM? x)², v(x) = k2(1 +iM? x), w(x) = k3(1− iM?²x),

(k) T`QpB/2b 2t+i bQHmiBQMb iQ i?2 j@bT2+B2b GQiF@oQHi2`` bvbi2K,

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

d1u^′′+ θu^′+ u(σ1− c¹¹u− c¹²v− c¹³w) = 0, x∈ R, d2v^′′+ θv^′+ v(σ2− c²¹u− c²²v− c²³w) = 0, x∈ R, d3w^′′+ θw^′+ w(σ3− c³¹u− c³²v− c³³w) = 0, x∈ R, (u, v, w)(−∞) ='

σ1

c11, 0, 0(

, (u, v, w)(+∞) = ' 0,_c^σ2

22, 0(

U8XRXRV

7Q` k1 = ^σ₄- k2 = ^σ₂- d1 = d2 = d3 = c11= c22 = c33= 1- σ1 = σ2 = σ3 = σ-

c21= 3c23− 1

σ(−1 + c23), c12= −8 − 3σ + c²³(3σ− 24)

σ(1− c23) , c13= (σ− 24)(c²³− 1)

16 ,

c32= 2(−σ − 8c23+ σc23)

σ(−1 + c²³) , c31= 8(−1 + 3c23)

σ(−1 + c²³), k3 = 4

−1 + c²³, θ = −4 + σ + 20c23− σc23

2(−1 + c23)

M/

16c13

−1 + c13

< σ < −8 + 24c13

−1 + c13

- B7 1 < c13≤ 3,

24 < σ < −8 + 24c13

−1 + c¹³ - B7 c13> 3.

hF2 σ = 28 M/ c23 = 4 7Q` 2tKTH2- i?2M c21 = ²²₂₁- c12 = ³⁷₂₁- c13 = ³₄- c32 = ²⁶₂₁-

kd

c31= ²²₂₁- M/ θ = −⁴₃X h?mb- U8XRXRV #2+QK2b

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

u^′′+ θu^′+ u(28− u − ³⁷₂₁v −³₄w) = 0, x∈ R, v^′′+ θv^′ + v(28−²²₂₁u− v − 4w) = 0, x ∈ R, w^′′+ θw^′+ w(28− ²²₂₁u−²⁶₂₁v− w) = 0, x ∈ R, (u, v, w)(−∞) = (1, 0, 0) , (u, v, w)(+∞) = (0, 1, 0) .

U8XRXkV

M/ i?2 2t+i bQHmiBQM Bb

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

u(x) = 7(1− iM? x)², v(x) = 14(1 +iM? x), w(x) = ⁴₃(1− iM?²x).

HbQ- r2 +M +?QQb2 u1 = u = 28- u₁ = u = ²⁸₂₂^·21- u2 = v = 28- u₂ = v = ²⁸₃₇^·21- u3 = w = ^28·4₃ - M/ u₃ = w = 7 BM h?2Q`2K 2.5 M/ h?2Q`2K 2.6X hF2 α1 = 1- α2 = ¹₅- M/ α3 = ¹₂ BM h?2Q`2K 2.5 M/ h?2Q`2K 2.6 7Q` 2tKTH2- r2 ?p2 i?2 7QHHQrBM; 2biBKi2b,

KBM)

28· 21 22 , 1

5 · 28· 21 37 , 1

2· 7

*

≤ u +1 5v +1

2w≤ Kt )

28, 1

5· 28, 1

2 ·28· 4 3

* ,

Q`

1

5 · 28· 21

37 ≤ u + 1 5v + 1

2w≤ 28.

h?2 mTT2` #QmM/ M/ HQr2` #QmM/ `2 b?QrM BM 6B;m`2 RRX

k3

−6 −4 −2 0 2 4 6 0

5 10 15 20 25 30

6B;m`2 RR, #Hm2 +m`p2, mc ;`22M +m`p2, pc TBMF +m`p2, rc #H+F +m`p2, u + ¹₅v + ¹₂wc

`2/ HBM2, mTT2` M/ HQr2` #QmM/b

8Xk M 1tKTH2 Q7 AKT`Qp2/ hM;2Mi GBM2 J2i?Q/

"v i?2 Mbix i?i u^′ = 8^m

i=0

aiuⁱ M/ v^′ = 8ⁿ

i=0

biuⁱ `2 #Qi? TQHvMQKBHb Q7 u- (3) T`QpB/2b 2t+i bQHmiBQMb Q7 URXyXdV mM/2` i?2 T`K2i2` bbmKTiBQM,

d = 3 a2

√σ, a1√

σ = 2 + 5√ σ 3 − a²,

√σ a₂ <√

σ < a1√

σ,θ = −2 + a√ ² 2a2

.

AM T`iB+mH`- 7Q` d = ²⁹₅- σ = 1- a1 = ²⁶₁₅ M/ a2 = ²⁹₁₅-

u(x) = 1 4

%

1− iM?

% x

√24

&&2

v(x) = 1 2

%

1 +iM?%

√x 24

&&

Bb M 2t+i bQHmiBQMX

hF2 α = 2 M/ β = ¹₃ BM h?2Q`2K 4.1 7Q` 2tKTH2X aBM+2

−α

αa1+^β_dσ(a2− 1)(≈ −0.5) < −α

β (= −6) < −α(a¹− 1) − ^β_dσa2 β

dσ (≈ −27.5),

kN

i?Bb Bb  iM;2Mi +b2- i?2M λ1 b?QmH/ #2 iF2M b

λ1 = αu0+β−(αa1u0+ ^β_dσ(a2u0− 1)) +=

(αa1u0+^β_dσ(a2u0− 1))²− 4^αβ_d σu0(u0− 1) 2^β_dσ

BM h?2Q`2K 4.1X A7 r2 +?QQb2 u0 = ^{−(B−EG)−}

√(B−EG)²−4(A−DG)(C−F G)

2(A−DG) - i?2M ^dv_du(u0)(≈

−56.3) ̸= ^−α_β (= −6)c r?BH2 B7 i?2 #`M+? Q7 u0 Bb +?Qb2M b

u₀ = −(B − EG) +;

(B− EG)² − 4(A − DG)(C − F G)

2(A− DG) , U8XkXRV

i?2M _du^dv(u0) = −6 = ^−α_β X h?mb- r2 +?QQb2 u0 b U8XkXRV BM i?2 2tT`2bbBQM Q7 λ1X

*QMb2[m2MiHv- i?2 HQr2` #QmM/ BM i?2 BKT`Qp2/ iM;2Mi HBM2 K2i?Q/ h?2Q`2K 4.1

#2+QK2b

2u(x) + 1

3v(x)≥ λ1

5

29 ≈ 0.05,

r?B+? Bb b?QrM BM 6B;m`2 RkX >Qr2p2`- 7`QK i?2 Q`B;BMH L@#``B2` KtBKmK T`BM+BTH2 h?2Q`2K 2.1- i?2 HQr2` #QmM/ Bb

KBM) 2· 15

29, 1 3· 15

26

* KBM>1, ²⁹₅ ? Kt>

1, ²⁹₅? = 5· 5

26· 29 ≈ 0.03,

r?B+? Bb rQ`b2 i?M i?2 HQr2` #QmM/ Q#iBM2/ pB i?2 BKT`Qp2/ iM;2Mi HBM2 K2i?Q/X

−300 −20 −10 0 10 20 30

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

6B;m`2 Rk, #Hm2 +m`p2, mc ;`22M +m`p2, pc #H+F +m`p2, 2u + <sup>1</sup><sub>3</sub>vc `2/ HBM2, HQr2` #QmM/

jy

e *QM+HmbBQM M/ 6mim`2 aim/B2b

6Q` QM2@/BK2MbBQMH KmHiB@bT2+B2b /BzmbBp2 +QKT2iBiBp2 GQiF@oQHi2`` bvbi2Kb- i?2 L@#``B2` KtBKmK T`BM+BTH2 biBHH T`QpB/2b  T`BQ`B 2biBKi2b 7Q` i?2 iQiH /2MbBiv Q7 i`p2HBM; rp2 bQHmiBQMbX b  +Q`QHH`v- MQM2tBbi2M+2b Q7 i`p2HBM; rp2 bQHmiBQMb BM QM2@/BK2MbBQMH /BzmbBp2 GQiF@oQHi2`` bvbi2K Q7 KmHiBTH2 +QKT2iBM; bT2+B2b `Bb2X 6m`i?2`KQ`2- i?2 BKT`Qp2/ iM;2Mi HBM2 K2i?Q/ K2HBQ`i2b i?2 HQr2` #QmM/ 2tTHB+@

BiHvX

PM2 Q7 i?2 +`m+BH bbmKTiBQMb Bb i?2 mMB7Q`KBiv Q7 /Bz2`2Mi bT2+B2b rp2b- BM #Qi?

p2HQ+Biv M/ /B`2+iBQMX "b2/ QM i?Bb ?vTQi?2bBb- r2 +M 2bBHv +?QQb2 i?2 `272`2M+2 +QQ`/BMi2b b i?2B` rp27`QMib- r?BH2 /B{+mHiB2b `Bb2 r?2M i?2 rp2b ;Q2b BM p`BQmb bT22/ Q` /B`2+iBQMX MQi?2` BMi2`2biBM; M/ T`+iB+H [m2biBQM Bb i?i r?2i?2` i?2`2

`2 bQK2 `2H2pMi `2bmHib 7Q` i?2 irQ@ Q` ?B;?2`@/BK2MbBQMH +b2X

d TT2M/Bt, JBMBKH qp2 aT22/

AM i?Bb b2+iBQM- r2 }`bi BMp2biB;i2 i?2 KBMBKH rp2 bT22/ Q7 i?2 6Bb?2`@EQHKQ;Q`Qp 2[miBQM URXyXkV #v T?b2 THM2 MHvbBb b BM *?Ti2` Rj Q7 (e)- M/ i?2M TTHv i?2 bK2 TT`Q+? iQ i?2 GQiF@oQHi2`` bvbi2KbX

*QMbB/2` i`p2HBM; rp2 bQHmiBQMb iQ i?2 6Bb?2`@EQHQKQ;`Qz 2[miBQM URXyXkV rBi?

i?2 #QmM/`v +QM/BiBQM u(−∞) = _c^σ₁₁¹, u(+∞) = 0, r2 ?p2

⎧⎪

⎨

⎪⎩

d₁u^′′+ θu^′+ u(σ₁− c11u) = 0, x∈ R, u(−∞) = _c^σ₁₁¹, u(+∞) = 0.

UdXyXRV

jR

hQ #2 bT2+B}+ r2 bbmK2 θ > 0X .2MQi2 i?2 }`bi /2`BpiBp2 u^′ #v U- i?2M

⎛

⎝u U

⎞

⎠

′

=

⎛

⎝ U

1

d1(−θU − u(σ¹− c¹¹u))

⎞

⎠ . UdXyXkV

`QmM/ i?2 bBM;mH`Biv (u, U) = (0, 0)-

⎛

⎝u U

⎞

⎠

′

≈

⎛

⎝ U

1

d1(−θU − σ1u)

⎞

⎠ =

⎛

⎝ 0 1

−σ1

d1

−θd1

⎞

⎠

⎛

⎝u U

⎞

⎠ . UdXyXjV

h?2 2B;2MpHm2b `2

λ =

−_d^θ₁ ±D'

θ d1

(2

− 4^σ_d¹₁

2 .

h?2M- B7 '

θ d1

(2

< 4^σ_d¹

1- i?2 2B;2MpHm2b rQmH/ #2 +QKTH2t MmK#2`b- M/ i?2M (u, U) rQmH/ #2  bi#H2 bTB`H M2` (0, 0)X h?Bb pBQHi2b i?2 MQM@M2;iBpBiv Q7 uX h?2`27Q`2- 'θ

d1

(2

≥ 4^σ_d1¹- Q` r2 ?p2  HQr2` #QmM/ 7Q` i?2 rp2 p2HQ+Biv,

θ ≥ 2;

σ1d1. UdXyX9V

LQi2 i?i- BM i?Bb +b2- i?2 2B;2MpHm2b `2 `2H M/ MQM@TQbBiBp2- M/ (u, U) Bb  bi#H2 MQ/2 M2` (0, 0)X PM i?2 Qi?2` ?M/- i i?2 Qi?2` bBM;mH`Biv (u, U) = (_c^σ₁₁¹, 0)-

⎛

⎝u U

⎞

⎠

′

≈

⎛

⎝ U

1

d1(−θU + σ1u)

⎞

⎠ =

⎛

⎝0 1

σ1

d1

−θd1

⎞

⎠

⎛

⎝u U

⎞

⎠ . UdXyX8V

h?2 2B;2MpHm2b `2

λ =

−_d^θ₁ ±D'

θ d1

(2

+ 4^σ_d¹₁

2 ,

r?B+? `2 `2H M/ ?p2 /Bz2`2Mi bB;Mb- ?2M+2 Bi Kmbi #2  b//H2 TQBMiX 6m`i?2`KQ`2- 7Q` i?2 TQbBiBp2 2B;2MpHm2- u+ M/ U+ Kmbi ?p2 i?2 bK2 bB;M BM i?2 +Q``2bTQM/BM;

2B;2Mp2+iQ`

⎛

⎝u+

U+

⎞

⎠- r?BH2 7Q` i?2 M2;iBp2 2B;2MpHm2- u<sub>−</sub> M/ U<sub>−</sub> Kmbi ?p2 /Bz2`2Mi

jk